
We note that the norm constraint here has dimension \(c\)  and \(\mathbf {w}_1\)  has dimension \(sd\) , so by [1]}, [2]} this strong duality result from Lemma REF  requires that \(m^* \le n\min \lbrace sd, c\rbrace \) . We introduce hyperplane arrangements \(\mathbf {D}_j\)  and enumerate over all of them, yielding
\(p_{FN}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^*(\mathbf {V}_i, \mathbf {Y}_i) \nonumber \\\mathrm {s.t.}~&\max _{\begin{array}{c}\Vert \mathbf {w}_1\Vert _2 \le 1 \\ j \in [P] \\ \mathbf {K}_j \mathbf {w}_1 \ge 0\end{array}} \Vert \sum _{i=1}^n \mathbf {V}_i^\top \mathbf {D}_j^{(i)} \mathrm {circ}({\mathbf {X}}_i) \mathbf {w}_1\Vert _2 \le \beta .\) 
